By an analytic family of algebraic cycles \(Z_s,s\in (\mathbb {C},0)\)  in \(X\)  we mean the following. We take an analytic variety \(Z\) , holomorphic maps \(f:Z\rightarrow (\mathbb {C},0)\)  and \(g: Z\rightarrow X\)  such that \(f\)  is proper and for all \(s\in (\mathbb {C},0)\) , \(g\)  restricted to \(Z_s:=f^{-1}(s)\)  is an injection.
The Gelfand-Leray form \(\frac{\omega }{ds}\)  in our context is only defined for \(\omega \in F^2H^3_{\rm dR}(X)=H^{30}\oplus H^{21}\) , for which the pull-back of \(\omega \)  to the two dimensional variety \(Z\)  by \(f\)  can be divided by \(df\) .
Similar to the case of Abel-Jacobi map, we also use the integration over the path of homology between two algebraic cycles. According to [1]} this goes back at least to [2]} which in turn must be inspired by Picard's intensive study of two dimensional integrals in [3]}. The proof of Theorem REF  actually implies that the Abel-Jacobi map attached to \(Z_s,s\in (\mathbb {C},0)\)  is identically zero, however, this is much weaker statement than the vanishing REF , see Section . The integration over algebraic cycles as in (REF ) appears first in Deligne's study of absolute Hodge cycles, see [4]}, and further applications of this in the study of Hodge loci has been initiated in [5]} [6]}.
